A concave mirror of focal length $f_1$ is placed at a distance of $d$ from a convex lens of focal length $f_2$

Question

A concave mirror of focal length $f_1$ is placed at a distance of $d$ from a convex lens of focal length $f_2$. A beam of light coming from infinity and falling on this convex lens concave mirror combination returns to infinity. The distance $d$ must be equal to:

Accepted Answer

1.  **Initial Image Formation:** A beam of light coming from infinity falls on the convex lens. The lens will converge these rays at its principal focus. Thus, the first image is formed at a distance $f_2$ from the convex lens.
2.  **Condition for Returning to Infinity:** For the final beam to return to infinity after reflecting from the mirror and passing through the lens again, the rays must retrace their original path. 
3.  **Retracing Condition for Mirror:** Light rays retrace their path when they strike a spherical mirror normally. This happens when the incident rays are directed towards the center of curvature of the mirror.
4.  **Relating Distances:** The image formed by the convex lens must lie at the center of curvature of the concave mirror. The distance of the center of curvature from the pole of the mirror is $R = 2|f_1|$.
5.  **Sign Convention:** According to standard sign convention, the focal length of a concave mirror is negative. Therefore, its magnitude is $|f_1| = -f_1$.
6.  **Total Distance Calculation:** The total separation $d$ between the lens and the mirror is the sum of the image distance from the lens and the radius of curvature of the mirror. 
$d = f_2 + 2|f_1| = f_2 + 2(-f_1) = -2f_1 + f_2$.

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